| L(s) = 1 | − 2·2-s + 2·4-s − 4·5-s − 4·7-s − 4·8-s − 3·9-s + 8·10-s + 2·11-s − 10·13-s + 8·14-s + 8·16-s + 2·17-s + 6·18-s − 2·19-s − 8·20-s − 4·22-s − 4·23-s + 5·25-s + 20·26-s − 8·28-s − 6·29-s + 8·31-s − 8·32-s − 4·34-s + 16·35-s − 6·36-s − 4·37-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 1.78·5-s − 1.51·7-s − 1.41·8-s − 9-s + 2.52·10-s + 0.603·11-s − 2.77·13-s + 2.13·14-s + 2·16-s + 0.485·17-s + 1.41·18-s − 0.458·19-s − 1.78·20-s − 0.852·22-s − 0.834·23-s + 25-s + 3.92·26-s − 1.51·28-s − 1.11·29-s + 1.43·31-s − 1.41·32-s − 0.685·34-s + 2.70·35-s − 36-s − 0.657·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34969 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34969 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 17 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 + 10 T + 48 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 47 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 75 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 75 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 82 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 4 T + 143 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 18 T + 200 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T + 140 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 155 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 28 T + 387 T^{2} - 28 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.15298367326208886726905911068, −11.82544775901667069597481524111, −11.49601174353460503855521245199, −10.87411142962362185284498662499, −9.935298124902701929565144568000, −9.799025126566522876008329274843, −9.464952410661559787722018480792, −8.914592410369310008235797861060, −8.141491757253270606536216131037, −7.88215004616533155477921585091, −7.54030749408128946585418505802, −6.54530980216965145943041093145, −6.50059977075890955087394229485, −5.51037147574392570723562926327, −4.66459895469057469235584488660, −3.71599331035951613782649292105, −3.16665993080278484109464679253, −2.50911346007699238392301203647, 0, 0,
2.50911346007699238392301203647, 3.16665993080278484109464679253, 3.71599331035951613782649292105, 4.66459895469057469235584488660, 5.51037147574392570723562926327, 6.50059977075890955087394229485, 6.54530980216965145943041093145, 7.54030749408128946585418505802, 7.88215004616533155477921585091, 8.141491757253270606536216131037, 8.914592410369310008235797861060, 9.464952410661559787722018480792, 9.799025126566522876008329274843, 9.935298124902701929565144568000, 10.87411142962362185284498662499, 11.49601174353460503855521245199, 11.82544775901667069597481524111, 12.15298367326208886726905911068
